# Spinor norm interpretation for a specific orthogonal group?

I hate to admit I don't know the answer to this but a referee has asked me about it in a paper of mine so here goes. Let F be a p-adic field with p odd. Let Q by a quaternary quadratic form over F of Witt rank one. Consider the orthogonal group G=O(Q,F). Then [G,G] is the kernel of the spinor norm and is isomorphic to PSL(2,E) where E is a quadratic extension of F determined by Q. My question is whether there is a nice interpretion of the spinor norm for G? For example if we replace Q by a ternary form of Witt rank one then the commutator is PSL(2,F), SO(Q,F) is PGL(2,F) and the spinor norm is given by det on PGL(2,F). Anything similar in the four dimensional case?

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What do you mean by "nice"? I would say that the spinor norm is very nice in itself, so it's hard to come up with something even nicer... –  Tom De Medts Apr 26 '11 at 12:27
The referee asked whether there is a simple explanation of the spinor norm in the quaternary case as there is in the ternary case in terms of det, as explained above. To my knowledge there isn't. It certainly isn't in Omeara or Dieudonne, for example. Agreed the spinor norm is very nice to begin with. Thanks for your comment. –  mander Apr 26 '11 at 13:52